Basic Information

Invidia was associated with the evil eye. There was significant fear that other's jealousy could work a curse on those who had been prosperous or lucky. To protect against this, one would wear a Fascinus, which is an amulet or sculpture of a penis. In fact, when a victorious General was given a Roman Triumph, the Vestal Virgins would bear a huge phallus ahead of him to ward off (or soak up?) the envy. The the Romans, being the target of envy was every bit as sinful or unpleasant as being envious oneself. Thus the need to carry around a phallic pendant. I think the TV Tropes wiki calls this Values Dissonance.

The Romans conflated Invidia with Nemesis, Adrestia, and Phthonus. Honestly, though, there's not much sense to connections to the first two, who are more about vengeance and war, respectively1. Phthonus is a more fitting equivalence, but there's a tiny problem of Phthonus being male, and Invidia being female.

Invidia is probably best understood through the lens of the early Roman gods, the numina. At the beginning, the Roman gods weren't anthropomorphic. They lacked bodies and personalities, so they didn't do much of anything. It wasn't till after the Romans interacted with the Greeks and Etruscans that they incorporated the idea of myths, stories, and identities for their gods. So, the most important Roman deities were grafted on to the existing Greek gods of the Illiad. Minor Roman numena that weren't worshiped, such as Invidia, never got myths of their own because the Romans just didn't do that. Rather than being a person, Invidia is just the concept of jealousy, and the magic of the evil eye. This may also address the Phthonus/Invidia question - if they are merely the anthropomorphisation of a concept, as opposed to a defined entity, then jealousy personified as a man is Phthonus and Invidia if a woman2.

The name Invidia was also used (as an intentional classical reference) for personifications of Envy of the 7 Deadly Sins, especially in gothic art of the middle ages. Sometimes she's represented as an ugly and terribly thin hag carrying a venomous serpent, or having one continuously bite her heart.

Sources

Game and Story Use

Too much success on the part of the PCs brings Invidia after them.

The PCs side won the war. Hooray! But now the glaring eye of Invidia is upon those who commanded the war efforts and most benefited from them. They may have been triumphant on the battlefield, but dark magic might strike them down at home. Depending on where the PCs sit in the national hierarchy, this could be a disaster or an opportunity.

As a jealous or envious creature that can bring sexual disfunction, Invidia could be associated with a Abyzou, Gello, Mormo, Succubus, or even Lilith. Break out your game's succubus stats and evil eye rules and sick Invidia on your players.

Some amusement could be had by having protective powers embodied in phallic amulets. Sure, it's a little juvenile, but that's rarely stopped gamers. Perhaps you get +1 to saving throws vs magic if you hold your phallus high. *snicker*

Of course if you are too widely envied, an amulet might not be enough … you might need to build a tower or something.

This may be why notably hubristic societies build skyscrapers…

Not a whole lot of detail or material exists readily on the internet about Invidia. So this is a goddess the GM can shape to his or her desire, and the players would be hard-pressed to tell you you're wrong.

This would also be a good in-universe explanation for any game balancing mechanics that you wish to apply to your rules set - a streak of sucesses lead to jealousy and grant Invidia (or an appropriatey expy thereof) power over a character.

Footnotes

1. One possible connection may be that Nemesis also punished the sin of hubris (pride or arrogance that offends the gods), so associating her with a spirit of jealousy might come in there.

2. Note also that invocations of the Roman deities tended to acknowledge their ability to assume whatever form, gender and name best suited them.